Yes differentiation is primarily about using only a portion of the genome as you pointed out. However, differentiated cells possess a kind of "memory" which enables the daughter cells to continue the usage of the same set of genes. This "memory" is enabled by epigenetic mechanisms which leave inheritable "marks" on the DNA such as methylation (I won't explain further as this would take an entire book).
About the place: each part of the body can produce cells locally (except for some tissues). For example the epithelial cells in your lips will arise from your lips. What signals the division of cells in response to a signal such as lets say injury is again a broad question. One simple mechanism is this (not an independent mode but just one aspect of a complex process): cells generally divide but stop dividing if there is no free space i.e. the cell is surrounded by other cells. This is called lateral inhibition of growth.
I would suggest that you pick up some good cell biology and molecular genetics books to understand these processes.
- Molecular Biology of the Cell - Bruce Alberts et al.
- Molecular Cell Biology - Lodish and Baltimore
- Molecular Biology of the Gene - James Watson et al.
- Genes - Benjamin Lewin